Trigonometry Examples

y = 4 cos (3 π x + \frac{2}{5})

$y = 4 \cos (3 π x + \frac{2}{5})$

Step 1

Use the form

a cos (b x - c) + d

$a \cos (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

a = 4

$a = 4$

b = 3 π

$b = 3 π$

c = - \frac{2}{5}

$c = - \frac{2}{5}$

d = 0

$d = 0$

Step 2

Find the amplitude

| a |

$| a |$ .

Amplitude:

4

$4$

Step 3

Find the period of

4 cos (3 π x + \frac{2}{5})

$4 \cos (3 π x + \frac{2}{5})$ .

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Step 3.1

The period of the function can be calculated using

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$

Step 3.2

Replace

b

$b$ with

3 π

$3 π$ in the formula for period.

\frac{2 π}{| 3 π |}

$\frac{2 π}{| 3 π |}$

Step 3.3

3 π

$3 π$ is approximately

9.42477796

$9.42477796$ which is positive so remove the absolute value

\frac{2 π}{3 π}

$\frac{2 π}{3 π}$

Step 3.4

Cancel the common factor of

π

$π$ .

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Step 3.4.1

Cancel the common factor.

$\frac{2 π}{3 π}$

Step 3.4.2

Rewrite the expression.

$\frac{2}{3}$

Step 4

Find the phase shift using the formula

$\frac{c}{b}$ .

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Step 4.1

The phase shift of the function can be calculated from

$\frac{c}{b}$ .

Phase Shift:

$\frac{c}{b}$

Step 4.2

Replace the values of

$c$ and

$b$ in the equation for phase shift.

Phase Shift:

$\frac{- \frac{2}{5}}{3 π}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift:

$- \frac{2}{5} \cdot \frac{1}{3 π}$

Step 4.4

Multiply

$- \frac{2}{5} \cdot \frac{1}{3 π}$ .

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Step 4.4.1

Multiply

$\frac{1}{3 π}$ by

$\frac{2}{5}$ .

Phase Shift:

$- \frac{2}{3 π \cdot 5}$

Step 4.4.2

Multiply

$5$ by

$3$ .

Phase Shift:

$- \frac{2}{15 π}$

Phase Shift:

$- \frac{2}{15 π}$

Phase Shift:

$- \frac{2}{15 π}$

Step 5

List the properties of the trigonometric function.

Amplitude:

$4$

Period:

$\frac{2}{3}$

Phase Shift:

$- \frac{2}{15 π}$ (

$\frac{2}{15 π}$ to the left)

Vertical Shift: None

Step 6